- This week's guest video was pitched to me as a puzzle
and I couldn't solve it.
So please welcome physicist Jade Tan-Holmes
who's going to show you the same puzzle
and then solve it with maths.
Jade, take it away.
- Here's a puzzle for you.
You have a painting, a piece of string, and two pins stuck in a wall.
Can you hang the painting in such a way
so that if you remove either one of the pins
the painting falls?
The thing to focus on here is how to tie the string around the pins,
not any of the physics of the objects.
So if you tie the string in the normal way one would hang a painting,
this obviously won't work.
It's pretty easy to think of ways
we can tie the string so it falls when one of this pins is removed.
But the challenge is to make it fall if either one is removed.
Pause the video now if you'd like to try it for yourself.
So what we're working towards is
a way to tie the string around both pins as a whole but neither one individually.
So there are a few different ways to solve this puzzle.
But we're going to do the one which takes us through knot theory.
Knot theory is the mathematical study of tangled systems.
As this is a problem about tangling things together,
chances are knot theory will come in handy.
With puzzles like these it's sometimes hard
to know where to even begin.
A common technique mathematicians use
is to strip away everything but the most important features.
So, let's think about what we can get rid of.
What parts of the problem aren't providing us
with any useful information?
Let's look at the painting.
It's cool, but is it really telling us anything?
It's just joining the two ends of the string.
If we get rid of it and tie the string together
we can reframe the question as
can we hang the string so that if either one
of the pins is removed, the string will fall?
It's just a simpler version of the same question.
Sorry, fashionable chicken.
[chicken squawks]
What else is unnecessary info?
Let's take a look at these pins.
Basically nothing matters about them
except how they interact with the string
and the fact that they don't interact with each other.
We don't care about their positions, size, shape or colour.
We don't even care that they're pins.
Mathematically, it makes sense to want to deal
with just one type of object instead of two.
Imagine reading a story that's written in
both English and Chinese.
Sure, both languages make up the whole story
but it'd be much easier to work with
if it was all the same language.
This is exactly what we're going to do now.
Translate our problem all into the same language,
the language of knot theory,
by converting our pins into strings.
All the properties we care about are still there.
If we tie them shut they can't interact with each other
and they can trap our original string.
In knot theory a closed string is called a knot
and a group of entangled knots is called a link.
Now we can reframe the question again as
can you tie the three knots so that when together they're interlocked,
but if you remove one, the others fall apart?
So far we've gotten rid of a bunch of information,
but we haven't really isolated what's important.
What should be focus on if we want to solve the puzzle?
Well, there's really only one way a string can interact with another.
It can cross over them or it can cross under.
The crossings are the only things we have control over
and are the key to unlocking the puzzle.
The crossings basically determine one knot from another.
But the thing is, the same crossing can look very different.
This makes knots hard to reason about in an organised way.
It's like if you had all the components of a story
but had no idea what order they were meant to be in.
But if we organise the crossings
we can reason much more clearly about the knot.
Let's take this random link of two knots.
If I start on a random string, at a random point
and start following the string, drawing on my paper as I go,
we can keep track of all the crossings in an organised way.
We keep going until we come back to where we started.
We've just created something called a braid diagram.
Knot theorists use these to model all kinds of links.
The point of braid diagrams is to keep track of the crossings
while getting rid of all the unnecessary information distracting us.
In a braid diagram,
the lines representing our strings are called strands.
If we label the strand positions one, two, and three,
when the strand in position one crosses over the strand in position two
we can label this crossing X,
and when the strand is position one crosses under the strand in position two,
we'll call this inverse X,
as it basically undoes what X just did.
When the strand in position two crosses over the strand in position three
we'll call this Y.
And when the strand in position two crosses under the strand in position three,
we'll call this inverse Y.
This notation is called a braid group.
And actually everything we need to know about our problem is encoded in these letters.
The way these letters work is that the X and inverse X cancel
and the Y and inverse Y cancel.
But they only cancel out if they are next to each other.
So coming back to our problem
we want the knots to be interlocked as a whole
but none of them locked individually to another.
So if we think about this in terms of our braid group,
removing a knot is like removing all crossings
involving that knot.
If we remove the green string all the Xs will be removed.
And if we remove the pink string
all the Ys will be removed.
What we want is a configuration where as a whole nothing cancels.
But if we remove either term, everything cancels.
Let's see what we can do.
We've got three strings.
Our system is symmetric, so what we do first really doesn't matter.
Let's just start with an X.
Remember the pins can't interact with each other,
so we have no choice but to do another X or inverse X.
We don't want to have an X and an inverse X next to each other
because we don't want anything to cancel yet.
So we'll do another X crossing.
This is just like wrapping one string around the other.
We could keep going like this with the Xs but we need to include the other string,
so let's do some Ys.
Now because we want our terms to cancel out in the end,
that means every X needs an inverse X
and every Y needs an inverse Y.
If we do two inverse Y crossings now
they'll immediately cancel out the Ys we just did,
which isn't what we want.
So let's try two inverse Xs.
Now we can do two inverse Ys.
So let's take a look at where we're at.
If you remove all the Y terms,
the remaining X terms cancel each other out.
And if you remove all the X terms,
the remaining Y terms cancel each other out.
But when everything is together, nothing cancels.
That's exactly what we want.
Now let's work backwards to the original puzzle,
chicken and all.
We can use the braid diagram to model our link.
Let's see if it has the property we want.
That when all three of them are together they're locked,
but if you remove one the others fall apart.
Yup!
Now let's bring back the chicken and the pins.
Okay guys, this is it.
So now we're going to map our braid group notation
onto our system of pins.
So, remember from the braid group diagram
that basically a double X corresponded to wrapping one string around the other.
And a double inverse X corresponded to undoing that move.
So here we're going to say that
a double X corresponds to wrapping around a pin clockwise
and a double inverse X corresponds to undoing that move
which would just be wrapping around the pin counterclockwise.
So, double X.
Double inverse X.
Double Y.
Double inverse Y.
So let's do it.
Double X
Double inverse X, wrapping around counterclockwise.
Double inverse Y.
So now the moment of truth.
[drumroll]
[thudding]
BOTH: Mathematics!
So this definitely wasn't the easiest
or most straightforward way to solve this puzzle.
So, why did we do it this way?
Well, now we have this whole framework
for solving this problem for any number of pins easily.
If we want to add another pin to the puzzle
this would be a nightmare to brute force.
But we can just work backwards from our braid group,
and it's simple.
Today we used knot theory to solve a fun puzzle,
but it actually leads to some very fundamental mathematics.
We were searching for a way to wrap the string
around both pins as a whole, but neither one individually.
But more broadly we had to keep track of the relationships between smaller parts
of a bigger system.
This kind of connection has led to deep insights in many areas of science.
Just recently, people have been exploring the design of quantum computers
using particles that behave just like the braids that we saw in this video.
So if you want to win a Nobel prize someday
better start tying some knots!
- Thank you Jade! Go subscribe to Up and Atom.
I would recommend starting with her video on quantum tunnelling.
And next week, a video that may leave you breathless.